Let V and W be vector spaces over F, and let T:VW be a linear transformation. Then: (1) dimF(T(V)) < dimF(V) and (2) If T is an isomorphism, dimF(T(V)) = dimF(V). Let V be a vector space. If V has a finite set of generators, then we say that V is finite-dimensional and we define its dimension, denoted dimFV, to be the minimum possible number of elements in such a set. If V does not have a finite set of generators, then we say the V is infinite-dimensional, and we define dimFV to be Let W V. Then W is of subspace of V For all w,w' W, F, we have w - w' W, w W. If f:GH is a homomorphism. The kernel of f is ker(f) = {x G | f(x) = 1H}. Let G1 and G2 be groups. An isomorphism from G1 to G2 is an injection :G1G2 such that (a · b) = (a) · (b)  (a,b G1). Let f :GH be a homomorphism. Then f is an injection ker(f) = {1G}. Let G and H be groups. A function f :GH which satisfies f(g · g') = f(g) · f(g') for all g, g' G is called a homomorphism of G into H. The set of real numbers. A function f : AB is said to be surjective (or onto) if for every y B there exists x A such that f(x) = y.

### Linear Transformations

In this section we will again pursue the analogy between vector spaces, on the one hand, and groups and rings on the other, by defining an analogue of the homomorphisms of group theory and ring theory. In the case of vector spaces, the homomorphisms are called linear transformations (or linear operators).

Definition 1: Let V and W be vector spaces over the same field F. A linear transformation from V to W is a function T:VW such that for all v,v' V and all F, we have

(1)
T(v + v') = T(v) + T(v'),
(2)
T(v) = T(v).

Note that a linear transformation T:W is a group homomorphism from the additive group of V to the additive group of W. Thus, in particular, if v,v' V, then

(3)
T(v - v') = T(v) - T(v').

Let v1,...,vn V, 1,...,n F. By repeated application of (1) and (2), we see that if T:W is a linear transformation, then

(4)
T(1v1 + ... + vn) = 1T(v1) + ... + nT(vn).

Example 1: Let 1,...,n F. Then the mapping T:FnF1 defined by T((a1,...,an)) = 1a1 + ... + nan is a linear transformation.

Example 2: Let V and W be finite-dimensional vector spaces over the field F and let {e1,...,en}, {f1,...,fm} be bases for V and W, respectively. Let T:W be a linear transformation. We may describe the effect of T as follows: Since T(ei) W, we have

T(ei) = a1if1 + ... + amifm   (1 < i < n),

where aij F. The set {aij}, which consists of mn elements of F, completely determines T, For if v V, then

v = 1e1 + ... + nen,   i F.

Therefore, by (4),

T(v) = 1T(e1) + ... + nT(en)
(5)
= 1(a11f1 + ... + am1fm)+ ... +n(a1nf1 + ... + amnfm)
= (a111 + a1nn)f1 + ... + (am11 + ... + amnn)fn.

Thus, the image of any vector v with respect to T is completely determined by the set {aij}. Conversely, given any set with {aij} of mn elements of F, formula (5) defines a linear transformation of V into W.

Example 3: Let V = W = F[X], considered as a vector space over F. If

f = a0 + a1X + ... + anXn F[X],

let us define the formal derivative of f, denoted Df, by

Df = 1 · a1 + ... + n · anXn-1.

Then D(f + g) = Df + Dg and D(f) = Df for all f,g F[X], F. Thus, D:F[X]F[X] is a linear transformation.

Example 4: Let C[0,1] = the vector space over R of all continuous functions f:[0,1]R. If f C[0,1], set

T(f ) = f(x)dx.

Then, by the properties if the integral proved in calculus, we have

T(f + g) = ( f(x) + g(x))dx
= f(x)dx + g(x)dx
= T(f ) + T(g),
T(f ) = f(x)dx
= f(x)dx
= T(f )

for all f,g C[0,1], R. Thus, T:C[0,1]R1 is a linear transformation.

As in group theory, and ring theory let us define the kernel of the linear transformation T:W, denoted ker(T), by

(6)
ker(T) = {v V | T(v) = 0}.

Proposition 2: Let T:VW be a linear transformation.

(1) T(V) is a subspace of W.

(2) ker(T) is a subspace of V.

Proof: (1) Let w,w' T(V), F. Then there exist v,v' V such that w = T(v), w' = T(v'). Then, by (3),

w - w' = T(v) - T(v') = T(v - v') T(V).

since v - v' V. Similarly, by (2),

w = T(v) = T(v) T(V),

since v V. Thus, by Proposition 2 of the section on subspaces, T(V) is a subspace if W. (2) Let v,v' ker(T), F. Then by (3) and the fact that v,v' ker(T),

T(v - v') = T(v) - T(v')
= 0 - 0
= 0
v - v' ker(T).

And, by (2),

T(v) = T(v) = 0 = 0
v ker(T).

Thus, ker(T) is a subspace of V.

Let T:W be a linear transformation. Since T is a homomorphism of additive groups, we have, by Corollary 5 of the section on group homomorphisms, the following result.

Proposition 3: T is an isomorphism ker(T) = {0}.

In our discussion of groups we remarked that one of the fundamental goals of the theory of finite groups is to make a list of all nonisomorphic finite groups, this goal being beyond the present state of knowledge. In the case of vector spaces, however, the situation is far different. The following theorem gives a complete classification of all finite-dimensional vector spaces over a field F

Theorem 4: Let V be a finite-dimensional vector space over a field F and let n = dimFV. Then VFn.

Proof: Let {e1,...,en} be a basis of V. Then every vector v V can be expressed uniquely in the form

v = 1e1 + ... + nen.

Define the function T:Fn by

T(1e1 + ... + nen) = (1,...,n).

It is easy to check that T is a linear transformation and T is clearly surjective. If v ker(T), then

T(v) = 0 (1,...,n) = (0,...,0)
1 = 2 = ... = n = 0
v = 0.

Thus, by Proposition 3, T is an isomorphism. Therefore, VFn.

Theorem 5: Let V and W be vector spaces over F, and let T:VW be a linear transformation.

(2) If T is an isomorphism, dimF(T(V)) = dimF(V).

Proof: (1) Without loss of generality, let us assume that V is finite-dimensional, and let {e1,...,en} be a basis of V. Then

V = {1e1,...,nen | ai F},

so that

T(V) = {1T(e1) + ... + nT(en) | i F},

and thus {T(e1),...,T(en)} is a set of generators for T(V). In particular, dimF(T(V)) < n.

(2) Let the notation be as in the proof of (1). It suffices to show that {T(e1),...,T(en)} is a basis for T(V). Since we have already shown that this set generates T(V), it suffices to show that {T(e1),...,T(en)} is linearly independent. If 1,...,n F are such that

0 = 1T(e1) + ... + nT(en).

0 = T(1e1 + ... + nen)

0 = 1e1 + ... + nen   (since T is an isomorphism)

1 = ... = n = 0   ({e1,...,en} is linearly independent)

{T(e1),...,T(en)} is linearly independent.

Theorem 5 has an interesting application to the theory of simultaneous linear equations. Let us consider the following system of equations:

a11x1 + a12x2 + ... + a1mxm = b1
a21x1 + a22x2 + ... + a2mxm = b2
.
(7)
.
.
an1x1 + an2x2 + ... + anmxm = bn,

where aij, bi all belong to a given field F. A solution of the system (7) is an m-tuple (x1,...,xm) Fm for which Equations (7) hold. A given system may or may not have a solution. In what follows we will study homogeneous systems - that is, systems of the form (7), in which b1 = b2 = ... = bn = 0. Such a system always has at least one solution, the zero solution (x1,...,xm) = (0,...,0). The following result guarantees the existence of nonzero solutions - that is solutions for which at least one xi is nonzero.

Theorem 6: Let

a11x1 + a12x2 + ... + a1mxm = 0
a21x1 + a22x2 + ... + a2mxm = 0
.
(8)
.
.
an1x1 + an2x2 + ... + anmxm = 0,

be a homogeneous system with coefficients aij belonging to the field F. If n < m, then the system always has a nonzero solution.

Proof: Consider the linear transformation T:FmFn defined by T((x1,...,xm)) = (a11x1 + ... + a1mxm,...,an1x1 + ...+ anmxm Fn. Note that (x1,...,xm) is a solution of the system (8)

T((x1,...,xm)) = (0,...,0)
(x1,...,xm) ker(T).

Assume that (8) has only the zero solution. Then ker(T) = {0}. Therefore, T is an isomorphism. By Theorem 5,

(9)

However, since T(Fm) Fn, we see that dimF(T(Fm)) < n. Therefore, by (9), m < n, which contradicts the assumption that n < m. Thus, (8) has nonzero solutions.